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plus construction on presheaves

This is a subentry of sheaf about the plus-construction on presheaves. For other constructions called plus construction, see there.

Idea

The plus construction $(-{)}^{+}:\mathrm{PSh}(C)\to \mathrm{PSh}(C)$ on presheaves over a site $C$ is an operation that replaces a presheaf via local isomorphisms first by a separated presheaf and then by a sheaf.

$$\mathrm{PSh}(C)\stackrel{(-{)}^{+}}{\to }\mathrm{SepPSh}(C)\stackrel{(-{)}^{+}}{\to }\mathrm{Sh}(C).$$PSh(C) \stackrel{(-)^+}{\to} SepPSh(C) \stackrel{(-)^+}{\to} Sh(C) \,.

Notice that in terms of n-truncated morphisms, if presheaf is

• separated precisely if every descent morphism is (-1)-truncated, namely a monomorphism;

• a sheaf precisely if every descent morphism is (-2)-truncated, namely an equivalence.

In the context of (n,1)-topos theory, therefore, the plus-construction is applied $(n+1)$-times in a row. The second but last step makes an (n,1)-presheaf into a separated infinity-stack and then the last step into an actual (n,1)-sheaf. (See Lurie, section 6.5.3.)

Definition

Definition

Let $C$ be a small site equipped with a Grothendieck topology $J$, let $A:C^{\mathrm{op}}\to \mathrm{Set}$ be a functor. Then the plus construction (functor) $(-{)}^{+}:\mathrm{PSh}(C)\to \mathrm{PSh}(C)$, resp. the plus construction $A^{+}$ of $A\in \mathrm{PSh}(C)$ is defined by one of following equivalent descriptions:

1. $A^{+}:U\mapsto {\mathrm{colim}}_{(R\to U)\in J(U)}A(R)$ where $J(U)$ denotes the poset of $J$-covering sieves on $U$.

2. Let $A:C^{\mathrm{op}}\to \mathrm{Set}$ be a functor. Then for $U\in C^{\mathrm{op}}$we define $A^{+}(U)$ to be an equivalence class of pairs $(R,s)$ where $R\in J(U)$ and $s=(s_f\in A(\mathrm{dom}f)\mid f\in R)$ is a compatible family of elements of $A$ relative to $R$, and $(R,s)\sim (R^{\prime },s^{\prime })$ iff there is a $J$-covering sieve $R^{\prime \prime }\subseteq R\cap R^{\prime }$ on which the restrictions of $s$ and $s^{\prime }$ agree.

3. $A^{+}:U\mapsto {\mathrm{colim}}_{(V\hookrightarrow U)\in V}A(V)$ where $W$ denotes the class $W:=(f^{*}{)}^{-1}\mathrm{Core}(\mathrm{Sh}(C{)}_1)$ of those morphisms in $\mathrm{PSh}(C)$ which are sent to isomorphisms by the sheafification functor $f^{*}$ and the colimit is taken over all dense monomorphisms only.

Properties

Remark

1. $+:A\mapsto A^{+}$ is a functor.

2. $A^{+}$ is a functor.

3. $A^{+}$ is a separated presheaf.

4. If $A$ is separated then $A^{+}$ is a sheaf.

Related concepts

• sheafification

• (∞,1)-sheafification / ∞-stackification

References

Related entries: sheafification

A standard textbook reference in the context of 1-topos theory is:

• Peter Johnstone, Sketches of an elephant - a topos theory compendium, Section C.2.2, proof of proposition 2.2.6, p.551

Remarks on the plus-construction in (infinity,1)-topos theory is in section 6.5.3 of

• Jacob Lurie, Higher Topos Theory

Plus construction for presheaves in values in abelian categories is also called Heller-Rowe construction:

• Alex Heller, K. A. Rowe, On the category of sheaves Amer. J. Math. 84 1962 205–216, MR144341, doi